-5 Is A Root Of The Quadratic Equation $n^2 - 8n - 65 = 0$.What Is The Other Root Of The Equation?

Feb 26, 2025 by ADMIN 99 views

**Solving Quadratic Equations: Finding the Other Root**

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to find the other root of a quadratic equation given one of its roots. We will use the example of the quadratic equation $n^2 - 8n - 65 = 0$ and show how to find the other root using various methods.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. The roots of a quadratic equation are the values of $x$ that satisfy the equation.

The Given Quadratic Equation

The given quadratic equation is $n^2 - 8n - 65 = 0$ . We are told that $-5$ is a root of this equation. This means that when we substitute $n = -5$ into the equation, it should satisfy the equation.

Substituting the Given Root

Let's substitute $n = -5$ into the equation:

$(-5)^2 - 8(-5) - 65 = 0$

Expanding and simplifying the equation, we get:

$25 + 40 - 65 = 0$

$0 = 0$

This confirms that $-5$ is indeed a root of the equation.

Finding the Other Root

Now that we have confirmed that $-5$ is a root of the equation, we need to find the other root. There are several methods to find the other root, including factoring, using the quadratic formula, and using the sum and product of roots.

Method 1: Factoring

One way to find the other root is to factor the quadratic equation. We can start by finding two numbers whose product is $-65$ and whose sum is $-8$ . These numbers are $-13$ and $5$ , since $(-13)(5) = -65$ and $-13 + 5 = -8$ .

We can now factor the quadratic equation as:

$n^2 - 8n - 65 = (n - 13)(n + 5) = 0$

This tells us that either $(n - 13) = 0$ or $(n + 5) = 0$ . Solving for $n$ in each case, we get:

$n - 13 = 0 \Rightarrow n = 13$

$n + 5 = 0 \Rightarrow n = -5$

We already know that $n = -5$ is a root, so the other root is $n = 13$ .

Method 2: Quadratic Formula

Another way to find the other root is to use the quadratic formula. The quadratic formula is given by:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$ , $b = -8$ , and $c = -65$ . Plugging these values into the formula, we get:

$n = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-65)}}{2(1)}$

Simplifying the expression, we get:

$n = \frac{8 \pm \sqrt{64 + 260}}{2}$

$n = \frac{8 \pm \sqrt{324}}{2}$

$n = \frac{8 \pm 18}{2}$

This gives us two possible values for $n$ :

$n = \frac{8 + 18}{2} = 13$

$n = \frac{8 - 18}{2} = -5$

We already know that $n = -5$ is a root, so the other root is $n = 13$ .

Method 3: Sum and Product of Roots

The sum and product of roots of a quadratic equation are related to the coefficients of the equation. Specifically, the sum of the roots is equal to $-b/a$ , and the product of the roots is equal to $c/a$ . In this case, we know that the sum of the roots is $-(-8)/1 = 8$ , and the product of the roots is $-65/1 = -65$ .

We can use this information to find the other root. Let's call the other root $r$ . Then, we know that:

$r + (-5) = 8$

$r(-5) = -65$

Solving for $r$ in each case, we get:

$r = 8 - (-5) = 13$

$r = -65/(-5) = 13$

We already know that $n = -5$ is a root, so the other root is $n = 13$ .

Conclusion

In this article, we have shown how to find the other root of a quadratic equation given one of its roots. We used the example of the quadratic equation $n^2 - 8n - 65 = 0$ and showed how to find the other root using various methods, including factoring, using the quadratic formula, and using the sum and product of roots. We found that the other root is $n = 13$ .

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will answer some of the most frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and using the sum and product of roots. We will discuss each of these methods in more detail below.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, simplify the expression and solve for $x$ .

Q: What is the sum and product of roots?

A: The sum of the roots of a quadratic equation is equal to $-b/a$ , and the product of the roots is equal to $c/a$ .

Q: How do I use the sum and product of roots?

A: To use the sum and product of roots, you need to plug in the values of $a$ , $b$ , and $c$ into the formulas. Then, simplify the expression and solve for the roots.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q: Can I use the quadratic formula to solve a linear equation?

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is only used to solve quadratic equations.

Q: Can I use the sum and product of roots to solve a linear equation?

A: No, you cannot use the sum and product of roots to solve a linear equation. The sum and product of roots are only used to solve quadratic equations.

Q: What is the relationship between the roots of a quadratic equation and the coefficients of the equation?

A: The roots of a quadratic equation are related to the coefficients of the equation. Specifically, the sum of the roots is equal to $-b/a$ , and the product of the roots is equal to $c/a$ .

Q: Can I use the quadratic formula to solve a cubic equation?

A: No, you cannot use the quadratic formula to solve a cubic equation. The quadratic formula is only used to solve quadratic equations.

Q: Can I use the sum and product of roots to solve a cubic equation?

A: No, you cannot use the sum and product of roots to solve a cubic equation. The sum and product of roots are only used to solve quadratic equations.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We have discussed the quadratic formula, the sum and product of roots, and the relationship between the roots of a quadratic equation and the coefficients of the equation. We hope that this article has been helpful in answering your questions about quadratic equations.

Additional Resources

If you are looking for additional resources on quadratic equations, we recommend the following:

Khan Academy: Quadratic Equations
Mathway: Quadratic Equations
Wolfram Alpha: Quadratic Equations

These resources provide a wealth of information on quadratic equations, including tutorials, examples, and practice problems.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. We hope that this article has been helpful in answering your questions about quadratic equations. If you have any further questions or need additional help, please don't hesitate to ask.